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The Journal of Physical Chemistry A

Scalable Quantum-Classical Hybrid Algorithm for Excited States Based on Divide-and-Conquer Unitary Coupled-Cluster Linear-Response Theory Using Dynamical Polarizability

Takeshi Yoshikawa, Tomoya Takanashi, Hiromi Nakai

Published Jun 30, 2026• 0 citations
Publisher

Abstract

This work introduces a divide-and-conquer (DC) quantum linear-response framework for scalable excited-state simulations, in which excitation energies and oscillator strengths are extracted from the poles of the frequency-dependent dynamical polarizability. This feature naturally enables a fragmentation-based formulation, while retaining the ability to describe nonlocal excitations beyond predefined localization regions. The method, termed DC-qUCCSD-LR, builds upon the established self-consistent quantum linear-response (qLR) theory combined with the variational unitary coupled-cluster ansatz with single and double excitations. For linear hydrogen chains, n H 2, the DC-qUCCSD-LR method reproduces full configuration interaction (FCI) excitation energies while significantly reducing quantum resource requirements, achieving favorable scaling of O ( n 1.5 ) for gate counts and O ( n 2.2 ) for measurements with respect to the molecular size, n . These results demonstrate that the polarizability-based qUCC-LR framework and its DC extension provide an accurate and scalable foundation for quantum excited-state simulations.

DOI
10.1021/acs.jpca.6c00669

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This work introduces a divide-and-conquer (DC) quantum linear-response framework for scalable excited-state simulations, in which excitation energies and oscillator strengths are extracted from the poles of the frequency-dependent dynamical polarizability. This feature naturally enables a fragmentation-based formulation, while retaining the ability to describe nonlocal excitations beyond predefined localization regions. The method, termed DC-qUCCSD-LR, builds upon the established self-consistent quantum linear-response (qLR) theory combined with the variational unitary coupled-cluster ansatz with single and double excitations. For linear hydrogen chains, n H 2, the DC-qUCCSD-LR method reproduces full configuration interaction (FCI) excitation energies while significantly reducing quantum resource requirements, achieving favorable scaling of O ( n 1.5 ) for gate counts and O ( n 2.2 ) for measurements with respect to the molecular size, n . These results demonstrate that the polarizability-based qUCC-LR framework and its DC extension provide an accurate and scalable foundation for quantum excited-state simulations.
In this study, we propose the quantum algorithm based on the unitary coupled cluster linear response theory for excited-state calculations with single and double excitations, denoted as qUCCSD-LR. Instead of the standard eigenvalue-problem-based scheme, the algorithm utilizes the dynamical-polarizability-based scheme, where the pole relative to the frequency of an external electric field corresponds to an excited state. Numerical applications of the qUCCSD-LR method to H2 could reproduce the dynamical polarizabilities, excitation energies and oscillator strengths obtained by the standard CCSD-LR method. Furthermore, potential energy curves for the double bond rotation in the ground and excited states of C2H4 were accurately calculated by the proposed method.
Quantum computing presents a promising avenue for solving complex problems, particularly in quantum chemistry, where it could accelerate the computation of molecular properties and excited states. This work focuses on computing excitation energies with hybrid quantum-classical algorithms for near-term quantum devices, combining the quantum linear response (qLR) method with a polarizable embedding (PE) environment. We employ the self-consistent operator manifold of quantum linear response (q-sc-LR) on top of a unitary coupled cluster (UCC) wave function in combination with a Davidson solver. The latter removes the need to construct the entire electronic Hessian, improving computational efficiency when going toward larger molecules. We introduce a new superposition-state-based technique to compute Hessian-vector products and show that this approach is more resilient toward noise than our earlier gradient-based approach. We demonstrate the performance of the PE-UCCSD model on systems such as butadiene and para-nitroaniline in water and find that PE-UCCSD delivers comparable accuracy to classical PE-CCSD methods on such simple closed-shell systems. We also explore the challenges posed by hardware noise and propose simple error mitigation techniques to maintain accurate results on noisy quantum computers.
The variational quantum eigensolver (VQE) with shallow or constant-depth quantum circuits is one of the most pursued approaches in the noisy intermediate-scale quantum (NISQ) devices with incoherent errors. In this study, the divide-and-conquer (DC) linear scaling technique, which divides the entire system into several fragments, is applied to the VQE algorithm based on the unitary coupled cluster (UCC) method, denoted as DC-qUCC/VQE, to reduce the number of required qubits. The unitarity of the UCC ansatz that enables the evaluation of the total energy as well as various molecular properties as expectation values can be easily implemented on quantum devices because the quantum gates are unitary operators themselves. Based on this feature, the present DC-qUCC/VQE algorithm is designed to conserve the total number of electrons in the entire system using the density matrix evaluated on a quantum computer. Numerical assessments clarified that the energy errors of the DC-qUCC/VQE calculations decrease by using the constraint of the total number of electrons. Furthermore, the DC-qUCC/VQE algorithm could reduce the number of quantum gates and shows the possibility of decreasing incoherent errors.
Fragmentation and embedding schemes are of great importance when applying quantum-chemical calculations to more complex and attractive targets. The divide-and-conquer (DC)-based quantum-chemical model is a fragmentation scheme that can be connected to embedding schemes. This feature article explains several DC-based schemes developed by the authors over the last two decades, which was inspired by the pioneering study of DC self-consistent field (SCF) method by Yang and Lee ( J. Chem. Phys . 1995, 103, 5674–5678). First, the theoretical aspects of the DC-based SCF, electron correlation, excited-state, and nuclear orbital methods are described, followed by the two-component relativistic theory, quantum-mechanical molecular dynamics simulation, and the introduction of three programs, including DC-based schemes. Illustrative applications confirmed the accuracy and feasibility of the DC-based schemes.
Exploring the potential application of quantum computers in material design and drug discovery has attracted a lot of interest in the age of quantum computing. However, the quantum resource requirement for solving practical electronic structure problems are far beyond the capacity of near-term quantum devices. In this work, we integrate the divide-and-conquer (DC) approaches into the variational quantum eigensolver (VQE) for large-scale quantum computational chemistry simulations. Two popular divide-and-conquer schemes, including many-body expansion~(MBE) fragmentation theory and density matrix embedding theory~(DMET), are employed to divide complicated problems into many small parts that are easy to implement on near-term quantum computers. Pilot applications of these methods to systems consisting of tens of atoms are performed with adaptive VQE algorithms. This work should encourage further studies of using the philosophy of DC to solve electronic structure problems on quantum computers.
Variational quantum algorithms on bosonic quantum processors are an emerging paradigm for quantum chemistry calculations, exploiting the natural alignment between molecular structure and harmonic oscillator-based hardware. We introduce the qumode-based variational quantum deflation framework (QumVQD) for finding both electronic and vibrational excited state energies on qumode-based architectures. We validate the approach through electronic structure calculations on H2_{\text{2}}2​ and linear H4_{\text{4}}4​, where we introduce Hamming-weight filtering of the Fock basis to enforce particle number conservation and eliminate spurious eigenstates by reducing the required Hilbert space, which reduces the required number of qumodes in turn. We achieve agreement with full configuration interaction (FCI) using the STO-3G basis set within the chemical accuracy threshold at most points along the potential energy surfaces. Extending to the vibrational structure, we combine QumVQD with an existing Hamiltonian fragmentation approach based on Cartan subalgebra, allowing us to compute the vibrational eigenenergies of CO2_{\text{2}}2​ and H2_{\text{2}}2​S to spectroscopic accuracy with per-fragment circuits that scale as O(N)O(N)O(N) in single-qumode gates and O(N2)O(N^2)O(N2) in beam-splitter gates for NNN qumodes. For the case of CO2_{\text{2}}2​, we get total gate counts more than an order of magnitude smaller than those reported for qubit-based vibrational algorithms at this system size. These results demonstrate that bosonic quantum devices are a viable platform for excited-state quantum chemistry, particularly for vibrational problems where qubit-based methods incur substantial boson-to-qubit mapping overhead.